Suppose Fred knows all necessary truths and is at least as smart as the author of this post. Fred wants to know whether a proposition p is true. So Fred says: “I stipulate that P is the singleton set {p} and that S is the subset of all the members of P that are true.” But sets have their members essentially. So S is necessarily empty or necessarily non-empty. If S is necessarily empty, then Fred knows that, and if S is necessarily non-empty, then Fred knows that, too. Since Fred is at least as smart as the author of this post, if Fred knows that S is necessarily empty, he can figure out that therefore S is empty, and hence that all the propositions in P are false, and hence that p is not true. And if Fred knows that S is necessarily non-empty, then Fred can figure out that therefore S is non-empty, and hence that p is true. In either case, then, Fred can figure out whether p is true.

To make this pointed, note that those open theists who think that there are facts about the future that God doesn’t know tend to think that God knows all necessary truths.

Genuine question: Should the open theist be worried by your argument? Suppose she has to give up on the claim that God knows all necessary truths. How problematic is that?
The question is partly motivated by the following combination of views:
1. Open Theism.
2. Some necessary truths must be known empirically, if they are known at all. Or at least some necessary truths can’t be known solely by a priori means.
3. God knows all necessary truths that can be known a priori.
4. For many of the sets you talk about, whether the set is empty or not would need to be known empirically or in some non-a priori way if it is to be known at all.
No doubt some further refinement is needed. But is there anything wrong with something like the conjunction of 1-4?

Not sure if I’m missing something here, but here’s my first thought. If you know all necessary truths, does that mean that you know that the proposition Iâm entertaining right now is true? (Suppose that, unbeknownst to you, the proposition Iâm entertaining right now is the claim that the square root of 625 is 25.) It seems that although you know that the square root of 625 is 25, you donât know that âthe proposition Bergmann is entertaining right now is trueâ. You do know of the proposition Iâm entertaining right now that itâs true, although you donât know which proposition Iâm entertaining, so you donât know that âthe proposition Bergmann is entertaining right now is trueâ (despite knowing all necessary truths). Likewise, you know that the empty set is empty, and so (assuming p is false) you know of S that it is empty but you donât know that S is the empty set, so you don’t know that “S is empty”.

Suppose p is contingently true. So it is contingently the case that the subset of all the members of P that are true is {p}. So it is contingently the case that Fred’s act of stipulation succeeded at picking out {p}. So—absent some further story—he does not know this.
But, you might say, his act of stipulation did succeed at picking out {p}. So ‘S’ refers to {p}. So S={p}. So, by the necessity of identity, necessarily S={p}, and so Fred knows that.
Let it be so. Still, he does not know that his act of stipulation succeeded at picking out {p}.
How can that be? For, consider:
1. He knows that ‘S’ refers to S
2. He knows that S={p}, so
3. So he knows that ‘S’ refers to {p}.
That, of course, is a familiar and problematic sort of argument. Let’s assume we are all Millians and we all accept the necessity of identity, so we all accept (2).
Then I’d suggest that either (1) is false: he does not know that ‘S’ refers to S, since it is contingent that ‘S’ refers to S, and for all he knows, ‘S’ might refer to the empty set instead, or (3) is true because ‘S’ essentially refers to what it does (words have their meanings essentially), and so essentially refers to S and essentially refers to {p}, and so Fred knows all that.
But if (3) is true for that reason, still he cannot infer from this that p is true, because he does not know whether or not the word he introduced is ‘S’ or a homophone of ‘S’ that instead refers to the empty set, since, of course, that is contingent upon the truth of p, and p is contingently true, and, absent some further story, he does not know p.
So, I conclude, he still does not know that his act of stipulation succeeded at picking out {p}, and so does not know that the word whose referent he stipulated refers to {p}. Without knowledge of that contingent fact, I do not see how he can infer that p is true.

In response to David and Michael’s very good comments, let me formalize a version of the argument.
1. Either <P is empty> is necessarily true or <P is empty> is necessarily false.
2. If <P is empty> is necessarily true, then Fred knows <P is empty>.
3. If <P is empty> is necessarily false, then Fred knows <P is non-empty>.
4. If Fred knows <P is empty>, then Fred knows that P is empty.
5. If Fred knows <P is non-empty>, then Fred knows that P is non-empty.
6. If Fred knows that P is empty, then Fred knows that p is false.
7. If Fred knows that P is non-empty, then Fred knows that p is true.
8. So, Fred knows that p is false or Fred knows that p is true.
This argument is valid. I think that Michael and David are implicitly questioning the disquotational principles (4) and (5), which follow from the general schema:
(A) (p)(If p is the proposition that s, then x knows p iff x knows that s)
or, equivalently:
(B) (p)(if p=<s>, then x knows p iff x knows that s).
Now I think (A) and (B) are very plausible.
But because of the Kripke puzzle, these schemata will get denied by people who take a Millian view of names (which is, alas, the majority view). For on a Millian view of names the proposition that London is beautiful is identical with the proposition that Londres is beautiful, but Pierre (let’s say) knows that London is beautiful and doesn’t know that Londres is beautiful.
So the open theist who doesn’t want to accept the conclusion of my argument (and some can–those who think there is no fact about future contingents) will question (A), (B), (4) and (5).
But there is still a cost for such an open theist. She will have to say that although:
(p)(If p is necessarily true, God knows p)
nonetheless there are sentences “s” such that (a) “s” expresses a necessary truth and (b) God does not know that s.
In general, I think the Millian’s best bet is to posit propositional guises. Thus, Pierre believes the same proposition under a “London” guise and does not believe it under a “Londres” guise. So our Millian open theist will say that while God knows all necessary truths, he doesn’t always know them under all guises.
But, interestingly, at this point the distinction between God’s knowledge of necessary and of contingent future truths gets blurred. Previously, we thought of the open theist as just saying that God knows all the necessary truths and none of the contingent future truths. But our Millian open theist has to admit that God knows quite a number of contingent future truths under some guise–which is just what she needs to say about God’s knowledge of necessary truths.
For suppose Jones will not mow the lawn tomorrow. Stipulate that “Jones*” is the name of Jones if Jones will not mow the lawn tomorrow and is the name of the Empire State Building if Jones will mow the lawn tomorrow. Then we all know that Jones* will not mow the lawn tomorrow, and so does God. But on a Millian view, the proposition expressed by “Jones* will not mow the lawn tomorrow” is the same as the proposition expressed by “Jones will not mow the lawn tomorrow”, since the two names co-refer, but the two sentences express it under different guises. So our Millian open theist has to say that the proposition that Jones will not mow the lawn tomorrow is known to God, albeit only under some guises.
So, in sum: Non-Millians should accept my argument. Millian open-theists will question it, but there is a cost.

Chris:
Open theists want to claim that God still knows , enough to qualify as omniscient.
For those open theists who think there are no future contingent truths, this is easy: they just say that God knows all truths.
But open theists, like PvI or Swinburne, who think that there are future contingent truths, have to work harder to be able to say that God is omniscient.
My argument is one of a family of arguments chipping away at the idea that such an open theist’s God would count as omniscient.
Two other members of this family are my argument that such a God would not know every proposition that it is logically possible that he knows, and Jon Kvanvig’s argument that there are propositions we know that such a God doesn’t know (e.g., that a very honest friend will freely refrain from cheating us).

Hi Alex,
I wonder whether your argument in this post is truly independent of the article you pointed us too. Assume that your other argument in RS and Kvanvig’s argument fail. There will be some future free contingent P that is unknowable by PvI’s lights. The necessary truth–that P is a member of the relevant set–would then be an unknowable necessary truth, because it is the sort of truth that can’t be known a priori. And then PvI-style reasoning kicks in to vindicate omniscience despite there being an unknowable necessary truth.

I might be missing something, but I don’t think Michael or I were relying on Millianism. (I was assuming Millianism because I thought otherwise your argument was a nonstarter. But I was not relying on guises…)
If you want to run this argument as a Fregean, you’ll need to say something about the sense associated with P (or, in your original example, S). If the sense is something like ‘the set {p} if p is true and otherwise the empty set”, then

is contingent, not necessary. The claim that sets have their members essentially does not entail the necessary truth of every proposition of the form . At best, it entails the de re necessity, of the set that happens to be the F, that necessarily it is F, and, I suppose, the necessity of the proposition that the F is empty, when ‘F’ expresses the essence of the relevant set.
Michael already gave a good example to make this point. Here is another: the set of flying pigs is empty. But, possibly, the set of flying pigs is not empty. So the proposition is only contingently true!

David,
Names refer rigidly. “The set {p} if p is true and otherwise the empty set” does not refer rigidly. A better rendering is “Dthe set {p} if p is true and otherwise the empty set”, where “dthe” is Kaplan’s rigidifying operator.
“the set of flying pigs is empty. But, possibly, the set of flying pigs is not empty. So the proposition is only contingently true!”
Right, the proposition <The set of flying pigs is empty> is contingently true. But the proposition <F is empty>, where “F” is in fact a name for the set of flying pigs, is necessary.
Which premise in my formalized version are you denying?
As for your original worry that “he does not know that ‘S’ refers to S, since it is contingent that ‘S’ refers to S”, the “since” seems to mark a non-sequitur. That “S” refers to S is a contingent a priori claim, like Kripke’s claim that the standard kilogram weighs a kilogram. Of course Fred knows that “S” refers to S, assuming Fred knows that “S” refers (and in the example, that’s guaranteed). That’s just disquotation, and he’s smart enough to disquote. Likewise, assuming we know there is a unique tallest man, we all know that “the tallest man” refers to the tallest man and “dthe tallest man” refers to the (or dthe) tallest man.
You also say that even given (3), “he does not know whether or not the word he introduced is ‘S’ or a homophone of ‘S’ that instead refers to the empty set”. Why not? He knows that the word he introduced is “S”–he introduced it, after all. And he knows that the word he introduced refers to S–that’s just disquotation–and since he knows all necessary truths, he also knows whether S is identical with the empty set, since that’s a necessary truth either way.
The point of the argument is that if you know all necessary truths and you know enough contingent a priori truths, you are in a position to find out all truths. But one can block this with guises and the like.

Michael:
“but you donât know that S is the empty set”
If S is the empty set, then it’s a necessary truth that S is the empty set, and so if you know all necessary truths, you know that S is the empty set. “S” isn’t a definite description like “the proposition that Bergmann is entertaining right now”; it is a name.
Likewise, I could give a name to the proposition that Bergmann is entertaining right now, say “Q”, and let’s suppose that, at least prima facie, unbeknownst to me Q is the proposition that the square root of 625 is 25.
Then I know Q. But it doesn’t follow that I know that the proposition that Bergmann is entertaining right now is true. Why not? Well, I do know that Q is the proposition that Bergmann is entertaining right now (that’s a contingent a priori claim). If I knew that Q is true, I could conclude that I know that the proposition that Bergmann is entertaining right now is true. But I don’t know that Q is true. I only know Q. So I can’t come to know that the proposition that Bergmann is entertaining right now is true.
But although I can’t, Fred can. For it is a necessary truth that Q is the proposition that the square root of 625 is 25. Propositions have their “content” (if one can even call it that, since propositions are contents!) essentially. And since it’s a necessary truth, Fred knows it. So Fred knows that Q is the proposition that the square root of 625 is 25. And Fred knows, since that’s contingent a priori, that Q is the proposition that Bergmann is entertaining right now. And if he’s moderately smart, he can figure out from that that the proposition that Bergmann is entertaining right now is true.
But I can’t. So there is no parody.

Here’s a (to my mind) slightly simpler argument for the same conclusion. I don’t know if the idea is essentially the same or not.
Suppose p. Then it is necessary that it is actually the case that p. Since Fred knows all necessary truths, he knows that it is actually the case that p. Also, since Fred is moderately smart, he knows that if it is actually the case that p, then p. So Fred can figure out that p.

Alex: My opening question was: âIf you know all necessary truths, does that mean that you know that the proposition Iâm entertaining right now is true?â It sounds like your answer is that if you are also moderately smart and you know of some name that it refers to the proposition Iâm entertaining right now (and I happen to be entertaining a necessary truth), then yes. (Have I understood you correctly?) If, despite being moderately smart, you don’t know of any name that it refers to the proposition that Iâm entertaining right now (so you’re not aware of yourself or anyone else giving a name to that proposition), is your answer still yes? Can the contingent act of assigning a name to it (or otherwise rigidly designating it) make the difference in whether you know that the proposition I’m entertaining right now is true?

Yeah, it’s simpler, but I think it uses somewhat more controversial assumptions, I think. The usual gloss on the rigidifying “Actually” operator is “At the actual world”. But this commits one to worlds.
And if the “Actually” is from ordinary language, then a case can be made that Actually(p) isn’t a necessary truth. Rather, one might take “Actually” to be a wide-scoping operator. I think that fits better with ordinary usage. Sam gets 90% on a test. Now take the proposition that he could have done better than he actually did. It’s not clear that this is a necessary truth. We could imagine someone saying that Sam could have done better than he actually did, and someone else saying: “Yes, and that wouldn’t have been true had Sam got 10% more.” And that wouldn’t be a counterpossible claim. But this isn’t knock-down.
Another way to run the argument is just to use the necessity of identity. Fred lets “A” be a name for the truth value of p. (Maybe the truth values are the abstract objects truth and falsity, or we can stipulate them to be the numbers 0 and 1.) Then Fred knows that A is, say, truth, since identity is necessary. So Fred knows that p is true. The commitment here is just to truth values, not even to propositions.

(At the grave risk of revealing my embarrassing ignorance) is any truth entailed by a necessary truth itself a necessary truth?
If so, wouldn’t Alex’s very cool argument entail that there are no contingent truths?

Michael has clarified to me by email that his questions to me were predicated on the assumption that like Fred, I know all necessary truths. So, second take.
Q1: “If you know all necessary truths, does that mean that you know that the proposition Iâm entertaining right now is true?” (Assuming that Michael is entertaining the proposition that the square root of 625 is 25.)
A: Yes. I can reason as follows. I stipulatively let p be the proposition Michael is entertaining right know. Then I know that p is the proposition that the square root of 625 is 25, since that’s a necessary truth, propositions having their “content” essentially. I also know that this proposition is true, since I know that the square root of 625 is 25. Finally, I know that p is the proposition Michael is entertaining right know, since given my stipulation, that’s a contingent a priori. Putting all this together, I will know that the proposition Michael is entertaining right now is true.
Q2: “If, despite being moderately smart, you don’t know of any name that it refers to the proposition that Iâm entertaining right now (so you’re not aware of yourself or anyone else giving a name to that proposition), is your answer still yes?”
A: Yes, but only because other variants of the same trick can be used. My variant used proper names. Jeff Russell (above) and Brian (in my personal blog’s comments) used the “actually” operator. I am not sure that works, for the reasons above. But even if that doesn’t work, I could just use demonstratives in place of proper names, since demonstratives refer rigidly. Thus, I ensure a context where “that proposition” consistently refers to Michael’s currently entertained proposition, and then it’s a contingent a priori truth that that proposition is the one that Michael is currently entertaining, and a necessary truth that that proposition is the proposition that the square root of 625 is 25.
Suppose I modify the question:
Q2b: Could I find out that the proposition he is entertaining is true if I made no stipulations, introduced no contexts, and if my critiques of the “actual” variant worked?
A: I don’t know. It depends on whether some other trick could be made to work. But let’s suppose it can’t for the sake of argument. Then, no.
Q3: “Can the contingent act of assigning a name to it (or otherwise rigidly designating it) make the difference in whether you know that the proposition I’m entertaining right now is true?”
Yes, that’s what the argument shows.
Now, there are perfectly ordinary cases of this. For instance, suppose that I know which proposition you’re entertaining, and it’s a complex mathematical proposition, and there is a very complex proof showing that the proposition is true. It could be that the proof is such that, given my contingent limitations, the only way I am going to be able to produce it is if I assign a name to the proposition you’re entertaining, so that I can then form various logical embeddings of it.
But the weirdness in the case of my argument and Michael’s version thereof is deeper. So, yes, this is weird.
(Maybe one should think of this argument as somewhat similar to the knowability “paradox” (I don’t see it as paradoxical myself).)

Just to be clear about what seems quite weird to me: Alex says âyesâ in answer to what he calls Q3. So we have a guy who knows all necessary truths and is otherwise smart but canât read minds, so he doesnât know what Iâm thinking. (Unbeknownst to him Iâm entertaining the proposition that the square root of 625 is 25.) Because he doesnât know what Iâm thinking, he doesnât know that âthe proposition Bergmann is entertaining right now is trueâ. However, he then assigns a name (say âSâ) to the proposition (whatever it is) that Iâm entertaining right now and, voila, he now knows, after a little thinking, that âthe proposition Bergmann is entertaining right now is trueâ.

Right.
I think the weirdness here is an indicator of the fact that it’s weird to suppose someone who knows all necessary truths, including the necessary a posteriori ones, without knowing all truths.
By knowing all necessary truths, Fred knows all world-indexed truths–he knows what happens in what world. The only thing more that he needs to know everything is to find a way to anchor his world-indexed truths to this world. And there are multiple tricks available for doing that by using a priori contingent knowledge. For instance, he might stipulate that “Jenny” is the name of the actual world, and since he then knows what truths hold at Jenny, as that’s necessary, and since he knows that Jenny is the actual world, as that’s contingent a priori, he then learns what truths hold at the actual world.
By the way, I originally came up with this argument for a non-theological case. Bayesianism requires that all necessary truths have probability one. This makes Bayesianism a theory about an idealized epistemic agent who knows all necessary truths, and at most a presentation of a regulative ideal for us. But the above argument shows it’s also a false theory about ideal epistemic agents. For such agents, if they’re moderately smart, isn’t going to be reasoning probabilistically. They’ll just going to be stipulating lots of names, and thereby finding out lots of stuff!
I am guessing that the Bayesian will retrench and say: We assign probability 1 to all a priori necessary truths, or maybe to all theorems of our favorite logic. But that’s costly. The notion of the a priori is dubious (I used the term in my arguments, but only expositively), and the choice of a logic is not sufficiently constrained.

That’s a really interesting argument. p is true iff. there is a singleton set S including that truth such that the the proposition expressed by ‘p is an element of S’ is necessarily true. But is it a necessary truth that p iff p is an element of S? No, it’s not. It’s merely stipulated that the set is called ‘S’ and so (at most) it’s apriori contingent that the biconditional is true. It is definitely not a necessary truth. You might have used the name ‘S*’ to name the set, for instance. So, Fred does not know that the biconditional, viz., p is true iff. there is a singleton set S such that the the proposition expressed by ‘p is an element of S’, is true, since it is not necessarily true. So Fred’s knowing that the right side of the biconditional is (say) false (since of course it is necessarily false or necessarily true) does not entail that he knows the left side is false. He would thereby know that the left side is false only if the biconditional itself were a necessary truth. But it isn’t. So Fred does not know the bioconditional is true. The same problem arises if you weaken the biconditional to a conditional.

Anonymous Too:
I’m not sure why S would have to be necessarily empty or necessarily non-empty.
For instance, let p be the proposition ‘Barack Obama is the POTUS’.
If we define:
P: = {p}
S: = {x: x is in P and y is true}
Then it seems to follow from the definitions:
p1: Necessarily, for all x, x is in P if and only if x is the proposition ‘Barack Obama is the POTUS’.
p2: Necessarily, for all x, x is in S if and only if x is the proposition ‘Barack Obama is the POTUS’ and x is true.
But the above entails that possibly, S is empty, and possibly, S is not empty, given that:
p2: Necessarily, for all x, x is in S if and only if x is the proposition ‘Barack Obama is the POTUS’ and x is true.
p3: Possibly, Barack Obama is the POTUS.
p4: Possibly, Barack Obama is not the POTUS.
p5: (Possibly Z and necessarily Y) entails (possibly (Z and Y)).
So, we get:
p6: Possibly, ((the proposition ‘Barack Obama is the POTUS’ is not true) AND (for all x, x is in S if and only if x is the proposition ‘Barack Obama is the POTUS’ and x is true)).
p7: Possibly, ((the proposition ‘Barack Obama is the POTUS’ is true) AND (for all x, x is in S if and only if x is the proposition ‘Barack Obama is the POTUS’ and x is true)).
And from that we get that possibly, S is empty, and possibly, S is non-empty.

Anonymous Too:
This depends on what your “:=” sign means. If it is an abbreviation-introduction sign, then p1 and p2 are true, but that’s not my argument, since in my argument “P” and “S” are names, not abbreviations. But if “:=” is a name-introduction sign, then p2 is false. Instead, all we have is the biconditional without the “Necessarily”.

@Alexander Pruss
I’d like to address the formalized version of your argument as well.
Let’s say that Fred stipulates that the word ‘Bob’ names the set whose only element is the proposition “Barack Obama is not the POTUS”, and that the word ‘Tom’ names the subset of Bob all of whose members are true.
The first premise of your formalized argument (substituting ‘S’ for ‘P’) is:
1. Either <S is empty> is necessarily true or <S is empty> is necessarily false.
Adapting this to the case of Obama, ‘Tom”, etc., we get:
1’: Either <Tom is empty> is necessarily true, or <Tom is empty> is necessarily false.
There are two ways of reading this:
1) If we let ‘Tom’ stand for the set that the word ‘Tom’ actually refers to, after Fred’s stipulation, then as it turns out, ‘Tom’ is the empty set, and clearly, the empty set is necessarily empty.
However, Fred does not know that the word ‘Tom’ actually refers to the empty set; that’s only contingently true.
More precisely, under this first reading, 1′ should be interpreted as:
1′(1): Either “The empty set is empty” is necessarily true, or “The empty set is empty” is necessarily false.
1′(1) is obviously true, and necessarily so, but of no help to Fred.
2) The second interpretation of 1′ would be:
1′ (2): Either “The set that the word ‘Tom’ refers to [whatever that set turns out to be, which varies depending on our stipulation, etc.] is empty” is necessarily true, or “The set that the word ‘Tom’ refers to [whatever that set turns out to be, which varies depending on our stipulation, etc.] is empty” is necessarily false.
However, 1′(2) is false, so it’s of no help to Fred, either.

Alex,
I may be missing something, but please bear with me. Since the proposition p is contingent it is generally the case that there is at least one possible world where it is true and one possible world where it is false. But then there is at least one possible world where S is non-empty, and one possible world where S is empty. Therefore it is not true that âS is necessarily empty or necessarily non-emptyâ as you claim. On the contrary it is generally the case that there is one possible world where S is empty and another where S is non-empty. – If this analysis is right then the trouble comes from your setting the argument in such a way that S refers to different things in different worlds, since how S is constructed depends on facts which obtain differently in different worlds. Which makes claims about necessary truths which involve S incoherent.
Anyway, hereâs a general counterargument. Assume that open theism is true, and that therefore God does not know truths about future free choices, such as, for example, whether I will choose eggs for breakfast tomorrow. Now since I am free it may be the case that I will choose eggs in all possible worlds â which makes the proposition âI will choose eggs for breakfast tomorrowâ necessarily true. Thus, on the assumption that open theism is true that particular necessary truth is not known to God. Therefore open theism entails that the proposition âGod knows all necessary truthsâ is perhaps false. But then one cannot use that proposition as a premise to prove that open theism is wrong.

The appeal to ‘dthat’ clarifies things for me, as does Jeff Russell’s simpler version of the argument (‘dthat’ seems to me like a sub-sentential version of what the ‘actually’ operator is supposed to be, in Jeff’s argument, putting aside how ‘actually’ actually behaves). I think I now agree that the weirdness is coming from supposing that Fred knows *all* the necessary truths.
So what is the plausible characterization of the constrained set of truths that God knows? Tradition suggests that God knows all the necessary truths in virtue of knowing the divine ideas, so it is tempting to suppose that what God knows are all the propositions made true by the essences of things. But that only pushes our problem back: are the essences of things prior to or posterior to what is actually the case? Kripke suggests that they are, at least in some cases, posterior…

“But then there is at least one possible world where S is non-empty, and one possible world where S is empty.”
Let’s suppose p is true in the actual world. And let’s suppose p is false in w.
Then S={p} in the actual world. What about in w? In w, S={p} still, since identity is necessary. But in w, it is false that S is the set of all true propositions in P.
The point here is that sets have their members essentially, and “S” isn’t an abbreviation for “the set of all true propositions in P”, but “S” is a name for whatever that set turns out to actually be, namely {p}.

David:
“So what is the plausible characterization of the constrained set of truths that God knows?”
I played with the idea that maybe the open theist can say that God knows all a priori necessary truths.
One problem with this is that what is a priori can differ from person to person, depending on conceptual resources.

Anonymous Too:
(I had to edit your post. To enter angle brackets, you need to use &gt; and &lt; codes.)
I reject both of your readings of 1′.
The first reading fails because even if Tom is the empty set, “The empty set is the empty set” is a poor translation of “Tom is the empty set”, just as “Bosphorus is Bosphorus” is a poor translation of “Hesperus is Bosphorus”.
The second reading fails because it has a different modal status from 1′. You can’t translate “Tom” as “The set that the word ‘Tom’ refers to”.
So how do I read “Tom is the empty set”? I read it as: Tom is the empty set. 🙂 I don’t know any way to paraphrase names away.

My logic-chopping skills aren’t good enough to figure out exactly where the problem is, but just on inspection it appears that there must be something wrong with this argument because it seems to show that all truths are necessary. Since S is either necessarily empty or necessarily non-empty, p must be either necessarily false or necessarily true (or else S wouldn’t have it members necessarily). But p was an arbitrary proposition, so it follows that all propositions are either necessarily true or necessarily false, i.e., there are no contingent propositions. This seems problematic.
Is there something wrong my thinking or does this argument prove too much?

“Since S is either necessarily empty or necessarily non-empty, p must be either necessarily false or necessarily true (or else S wouldn’t have it members necessarily).”
No, this doesn’t follow, since that S is empty does not entail that p is false (there is an a priori contingent implication here) and that S is non-empty does not entail that p is true.

Why doesn’t it entail that? You defined S to be “the subset of all the members of P that are true,” where P is the singleton set {p}.
If S is the empty set, then necessarily p is not a member of S. Since p is necessarily a member of P and S is the set of all members of P that are true, it follows that necessarily p is false.
Conversely, if S is non-empty, then necessarily at least one member of P is a member of S. Since P is just {p}, then it follows that necessarily, p is a member of S. Then we have that p is necessarily true.
Since these are the only two possibilities, it follows that either p is necessarily true or p is necessarily false.
Am I missing something here?

I didn’t define S as “the subset of all the members of P that are true”. I let “S” be a name for the subset of all the members of P that are true.
There is a difference.
Suppose I define a tilly as the tallest person in the world. Then that a tilly exists entails that a tilly is taller than any other person.
But suppose I let “Tilly” be a name for the tallest person on earth (whoever it may be). Then that Tilly exists does not entail that Tilly is taller than any other person, since in some worlds Tilly is in fact quite short, due to having received growth-retarding drugs as a kid.

“I didn’t define S as “the subset of all the members of P that are true”. I let “S” be a name for the subset of all the members of P that are true.”
All right, so if I understand this correctly you’re saying that the difference is that S is the name given in world w to the set whose members are just those members of P that are true in w. So the members of S are necessarily members of S because sets have their members essentially, but S is not necessarily the set of all members of P that are true because there is a possible world w* where p is not true (if p is true in w), or conversely p is true if p is not true in w. I take it that this is what you mean by the statement that the fact that S is empty does not entail that p is false (correct me if I am wrong about any of this).
However, it does seem like the fact that S is empty entails that p is false in w, since if p were true in w then the name “S” would not identify the subset of all members of P that are true in w, which is in contradiction to how we have stipulated the name “S.”
This may not make p a necessarily false proposition, but it does give us a very peculiar sort of inference. From the necessary statement “S is empty” it follows (if not necessarily then at least analytically) that p is false in w. But if p is necessarily or analytically false in w, wouldn’t it have to be the same in every possible world? Could the problem be that you need some kind of contingent a posteriori knowledge in order to know to which set the name “S” refers (and thus a being that knew all necessary truths would not be able to tell you whether S is empty or not because he would not know which set you’re talking about)?
Somebody much smarter than me needs to hash this out. My sense is that there’s definitely something fishy going on here though. I just don’t see how you’re going to be able to derive contingent facts from purely necessary priors and not have those facts themselves be necessary.

Anonymous:
“My sense is that there’s definitely something fishy going on here though.”
I have the same sense, but I can’t see what’s wrong with the argument.
“Could the problem be that you need some kind of contingent a posteriori knowledge in order to know to which set the name ‘S’ refers?”
I assume that by “which set” you mean “which of {} and {p}”. (There is another piece of knowledge that Fred needs to have. He needs to know that “S” refers to the set of all the truths in P. This is a genuine piece of contingent knowledge, but it is one that Fred unproblematically has by observing his own stipulation.)
It’s tempting to say the following: What the argument shows is that there are necessary truths that can only be known if you know some pretty substantive contingent truth. That doesn’t sound paradoxical.
But I actually think that the italicized claim is false. For consider the necessary truth that S is non-empty. (Supposing that p is in fact true.) Then presumably the substantive contingent truth that one needs to know, in order to know that S is non-empty, is p.
But if God necessarily knows all necessary truths, then it is false that one needs to know p to know that S is non-empty. For that S is non-empty is a necessary truth. In particular, that S is non-empty is true even in those worlds in which p is false. But since God necessarily knows all necessary truths, then in those worlds God also knows that S is non-empty. But in those worlds he doesn’t know p, since in those worlds p isn’t true, and no one can know a falsehood.
Perhaps, though, we can say that God’s knowledge that S is non-empty is grounded in the truth of p in those worlds in which p is true, but in those worlds in which p isn’t true, God’s knowledge is grounded in something else. But now we have the puzzle: Why doesn’t that “something else” ground God’s knowledge that S is non-empty in our world?
This is puzzling stuff.

“Why doesn’t that “something else” ground God’s knowledge that S is non-empty in our world?”
I’m also troubled by the italicized statement, but for a different reason.
I think the greater danger may be that there could be some necessary truth p and some world w for which w does not provide the contingent grounds necessary to derive p. After all, being contingent, there’s no reason to suppose that such grounds should exist in an arbitrary world.
The issue for me is that if there are necessary truths that are unknowable without knowing some substantive contingent truth, then it seems that we’ve undermined a major motivation for thinking that God would know every necessary truth. The reason why we would expect a God-like being to know every necessary truth, to my intuition, is that we can think of God as being “maximally intelligent” in the sense that there would not be any logical implications that escape his understanding. But if we accept that there are necessary truths that cannot be known without also knowing a substantive contingent truth, then there doesn’t seem to be a reason why God should be guaranteed to know these truths if he doesn’t also know all contingent truths.
Maybe it would make sense under this scheme to say that God knows all analytically true statements, but not all necessarily true ones?

“Maybe it would make sense under this scheme to say that God knows all analytically true statements, but not all necessarily true ones?”
That would require a workable analytic-synthetic distinction, and I am sceptical of one.

Jon Kvanvig essentially (i.e., in response to another variant of the puzzle) suggested to me that we take “S” to be a variable or schematic letter.
Here’s my way of working through one version of that solution, taking “S” to be a variable letter. Consider this tightening-up of my argument.
1. Let S be the subset of {p} consisting of all the truths.
2. Then either S is necessarily empty or S is necessarily non-empty.
3. If S is necessarily empty, then Fred knows that S is empty.
4. If S is necessarily non-empty, then Fred knows that S is non-empty.
and so on.
But this fails at steps 3 and 4, because the phrase “S is empty” does not express a proposition, since it’s a formula containing an unbound variable letter, whereas we only assumed that Fred knows all propositions that are necessarily true.
On this take, my previous formalization might have actually contained nonsense. We can’t talk of <S is empty>, because the angle bracket operator requires a sentence rather than an open formula as an argument.
Compare this line of thought:
a. Let p be any odd prime number.
b. Then p is not 1 or 2.
c. So, p is greater than or equal to 3.
d. So all odd primes are greater than or equal to 3.
This line of thought makes perfect sense. But we certainly don’t want to take “p” to be a name (in the ordinary sense; logicians can stipulate “name” as they wish). For if “p” were a name, we could ask which odd prime number it refers to, and that’s a nonsense question.
If we were to formalize a-d further, by using a formal language, a-c might be turned into a universal generalization subproof. Now the lines of a subproof do not function as assertions–they play some other linguistic role. And it is reasonable to say that they need not actually be sentences in the ordinary sense (again, logicians can stipulate as they wish) expressing propositions.